If you want to go, and think your kids are reasonably adaptable, I wouldn't hesitate - Rajasthan is fab! :)
We have taken dd (6) to India four times, now - DH is from India, and all his family are there, so we don't really have much choice. She has never been ill, and despite some concerns on my part each time, she has loved every moment of every trip.
I have lost count of how many times I have been myself, but I have never been ill either, apart from a common cold. If you are sensible about what you eat and drink, there is no real reason why you should get sick - happy to advise further if you want.
The climate in November will be good, so no worries on that score in Rajasthan - no idea about Goa or Kerala, though.
I suppose India is a bit of a culture shock when you go there for the first time, but it's all part of the rich tapestry of life, and I think kids take things in their stride much more easily than we might imagine. I probably wouldn't have taken dd at this age if I didn't have to, but I am so glad that she has had the experience. Each time, I wonder if she will be uncomfortable about stuff, and each time, she surprises me with how well she copes - she just takes it all in, and she has learned so much from her visits.
So, I would say go, and enjoy the adventure!